As an intro, I know how the numbers are represented, how to do it if I can calculate powers of the base, and then move between base $m$ to base $10$ to base $n$. I feel that this is overly "clunky" though, and would like to do it in such a way that the following conditions are met:

No need to calculate the powers of the base explicitly

No need for intermediate storage (i.e. no conversion to base ten required if base ten is not one of the bases)

I am pretty sure that the only operations that I strictly need to use are modulo, division and concatenation, but I can't seem to figure it out.

Confused. Did you just define $x$ to be two different things? How did I get $x$'s base $b$ representation?
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soandosFeb 20 '12 at 2:10

Oops I am thinking like a programmer. You repeat the procedure on $x//b$, where the $$//$$ means "integer divide (discard remainders)."
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ncmathsadistFeb 20 '12 at 2:16

Wait, how do I do this for a starting base $\neq10$?
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soandosFeb 20 '12 at 2:25

This procedure works regardless of base; however arithmetic must be executed in that base. If you come from the planet Tridigia where homonids have three fingers on each hand and work in Base 6, they would proceed in the same way, using base 6 arithmetic. This is base-invariant. Try the exercise of doing it in another base to convince yourself.
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ncmathsadistFeb 20 '12 at 2:27